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Intraspecific Trait Variation and its
Effects on Food Chains
Don DeAngelis
University of Miami
Coral Gables, Florida USA
Workshop on Nonlinear Equations in Population Biology
East China Normal University, Shanghai, China
May 25-27, 2013
Intraspecific Variation
1.
Traits such as skill at foraging and investment in antipredator defense may vary among individuals within a
species population.
2.
Traits such as the choice of what sort of habitat to utilize
can also vary among individuals of a population.
Here the effects of both types of variation are examined.
This intraspecific variation has implications for both fitness
strategies within a population and food web dynamics.
I. Intraspecific Variation in foraging ability,
predator avoidance, and other mortality risk
Intraspecific variation within populations has been shown to
be nearly ubiquitous in nature and to play an important role
in community dynamics:
•
D. I. Bolnick, P. Amarasekare, M. S. Araújo, R. Bürger, J. M. Levine, M. Novak, V. H. W. Rudolf,
S. J. Schreiber, M. C. Urban, and D. A. Vasseur, Why intraspecific trait variation matters in
community ecology, Trends Ecol. Evol,, 26:183-192 (2011).
•
D. I. Bolnick. R. Svanbäck, J. A. Fordyce, L. H. Yang, J. M. Davis, C. D. Hulsey, and M. L. Forister,
The ecology of individuals: incidence and implications of individual specialization. Amer.
Natur., 161:1-28 (2003).
•
M. Wolf, and F. J. Weissing, Animal personalities: consequences for ecology and evolution.
Trends in Ecol. Evol., 27:452-461 (2012).
Basic Questions
• How does this intraspecific variation relate to strategies
for fitness of individuals in a population?
• Given the existence of subpopulations having distinct
sets of traits or strategies within a population, and that
there is probably continuous switching of individuals
between these subpopulations affect these dynamics?
M
Predator
To investigate trait variation,
we consider a tri-trophic
chain of ‘resource R’,
‘consumer N’, and ‘predator
M’
f1N1M
f2N2M
m12N1
N1
Consumer 2
Consumer 1
The consumer is assumed to
have two phenotype
subpopulations (with
different strategies) and
there is some switching back
and forth between the two.
N2
m21N2
a1RN1
R
Resource
a2RN2
This chain resembles the
‘diamond-shaped’ chain that
has been studied before;
e.g.,
E. G. Noonburg and P. A.
Abrams, Transient dynamics
limit the effectiveness of
keystone predation in
bringing about coexistence,
Amer. Natur. 165:322-335.
(2005).
In this case the two
consumers are different
species, so there is no
movement between the two
consumer strategies.
M
f1N1M
f2N2M
N1
N2
a1RN1
a2RN2
R
A simple set of equations for this system is as follows:
dR
R

 rR1    a1 RN 1  a 2 RN 2
dt
 K
Resource
,
dN1
 ba1 RN 1  d1 N1  f1 N1 M  m12 N1  m21 N 2
dt
Consumer Phenotype 1
dN 2
 ba 2 RN 2  d 2 N 2  f 2 N 2 M  m12 N1  m21 N 2
dt
Consumer Phenotype 2
dM
 cf1 N1 M  cf 2 N 2 M  d m M
dt
Parameters ai , di, and fi may differ between the two
consumer phenotypes.
Predator
There are two equilibrium points for the resource and
consumers alone
d 
r 
1  1 
a1  Kba1 
R1* 
d1
ba1
N1*,1 
R2* 
d2
ba2
N 2*,1  0
N 2*,2 
N1*,2  0
r
a2
M 1*  0

d 
1  2 
 Kba2 
M 2*  0
However, the equilibrium below cannot exist, as only
one species can survive in this model of exploitative
competition. The better forager excludes the other.
R2* 
d2
ba2
N 2*,1 
r
a2

d 
1  2 
 Ka2 b 
N 2*,2 
r
a2

d 
1  2 
 Ka2 b 
M 2*  0
But at least one of the tri-trophic chains is assumed exist; i.e., the better
forager is poorer at evading the predator.
 ad 
R3*  K 1  1 m 
rcf 1 

N 3*,1 
 ad 
R4*  K 1  2 m 
rcf 2 

N 4*,1  0
dm
cf1
N 3*,2  0
N 4*,2 
dm
cf 2
M 3* 
ba1 K
d1
(
rcf

a
d
)

1
1
m
f1
rcf 12
(1)
M 4* 
ba2 K
d2
(
rcf

a
d
)

2
2
m
f1
rcf 22
(2)
Assume that (1) exists; that is, that
cf 1 N 1*,1  d m  0
This provides a path to the full system, if consumer 2 can invade; i.e., if
 ad
ba 2 R3*  d 2  f 2 M 3*  ba 2 K 1  1 m
rcf 1


 ad 
  d 2  ( f 2 / f1 )( ba1 K 1  1 m   d1 )  0
rcf 1 


This means that the poorer forager can now invade, because the predator
suppresses the better forager to some extent.
To obtain this solution we also make the assumption that the consumers
are in an Ideal Free Distribution at equilibrium, so that m12N1* = m21N2*.
This means that the individuals are distributed among the two strategies
such that changing will not improve their fitness. There is still switching,
but it is balanced.
We can find the interior equilibrium point.
R5* 
f 2 d1  f 1 d 2
b( a1 f 2  a2 f1 )
M 5* 
ba1  f 2 d1  f1d 2  d1

f1 b( a1 f 2  a2 f1 ) f1
N 5*,1 
a2 d m
f r [ Kb( a2 f1  a1 f 2 )  ( f 2 d1  f1d 2 )]
 2
c( a2 f1  a1 f 2 )
Kb( a2 f1  a1 f 2 )2
N 5*,2 
 a1d m
f r [ Kb( a2 f1  a1 f 2 )  ( f 2 d1  f1d 2 )]
 1
c( a2 f1  a1 f 2 )
Kb( a2 f1  a1 f 2 )2
M
f1N1M
f2N2M
E. G. Noonburg and P. A.
Abrams (2005) studied the
stability of the diamondshaped food web.
N1
N2
a1RN1
a2RN2
They found that local
stability of the interior
equilibrium point was
possible.
R
From Noonburg and Abrams ,
American Naturalist 2005
However, they found that
the system exhibits slowly
damped extreme
fluctuations, when the
second consumer is
introduced at small values.
This would likely lead to
extinction of one or more of
the species.
The question then is, what is the stability behavior of the
analogous model for two consumer phenotypes in which
continuous switching among the phenotypes occurs?
Analysis of the eigenvalues from the matrix

r *
R3    a1 R3*
 a2 R3*
0
K
ba1 N 3*,1
 m12  
m21
 f1 N 3*,1  0
ba2 N 3*,2
m12
m21    f 2 N 3*,2
0
cf1M 3*
cf 2 M 3*

shows that local stability is possible.
The eigenvalues can
be further studied as
a function of the
movement rate m12,
with m21 given by
m12 N1*/N2*.
For m12 small,
dynamics is
dominated by
complex conjugate
eigenvalues with small
negative real part.
As m12 is increased,
the absolute value of
the real part increases
and then the solution
bifurcates to two real
eigenvalues.
Simulations
For small m12 in the
two consumer
phenotype system,
slowly damped large
oscillations occur, as
in Noonburg and
Abrams (2005)
model of a diamondshaped web.
N1*
N2*
R*
For larger m12, the
transients damp more
rapidly.
N2*
N1*
R*
For still larger m12 the
long-term transient
ceases to oscillate and
damps monotonically
N2*
The remaining complex
conjugate roots change
little and continue to
cause very rapidly
damped oscillations, but
these are not serious.
N1*
R*
What are the properties of this system at equilibrium when certain
parameters, such as resource carrying capacity, K, and predator
mortality rate, dm ?
The equilibrium is written in a form to emphasize the dependence.
R5* 
f 2 d1  f 1 d 2
b( a1 f 2  a2 f1 )
M 5* 
ba1  f 2 d1  f1d 2  d1

f1 b( a1 f 2  a2 f1 ) f1
N 5*,1 
a2 d m
f r [ Kb( a2 f1  a1 f 2 )  ( f 2 d1  f1d 2 )]
 2
c( a2 f1  a1 f 2 )
Kb( a2 f1  a1 f 2 )2
N 5*,2 
 a1d m
f r [ Kb( a2 f1  a1 f 2 )  ( f 2 d1  f1d 2 )]
 1
c( a2 f1  a1 f 2 )
Kb( a2 f1  a1 f 2 )2
Let’s first review what happens in the single-consumer chain
,
dR
R

 rR 1    aRN
dt
 K
Resource
dN
 baRN  dN  fNM
dt
Consumer
dM
 cfNM  d m M
dt
Predator
with the equilibrium solution
 ad 
R*  K 1  m 
rcf 

N* 
dm
cf
M* 
baK
d
(
rcf

ad
)

m
f
rcf 2
Note that when K increases, R* increases and M* increases; N*
does not change – alternating effects of bottom-up control.
When dm increases, M* decreases, N* increases, and R* decreases –
cascading effects of top-down control.
However, in the two
consumer phenotype
model, when K
increases,
M*
R* and M* remain the
same and N1* + N2*
either increases or
decreases, the latter
in this case.
N1* + N2*
N2*
N1*
R*
In the case of two
consumer phenotypes,
when dm increases
R* and M* stay the
same, and N1* + N2*
either increases or
decreases.
Therefore, there is a
complete change in
the bottom-up and
top-down effects in a
food chain when
there are stably
coexisting phenotypes
of one consumer
species.
M*
N1* + N 2*
N1*
N 2*
R*
Conclusions
Trophic cascades have been the object of intense interest by ecologists over the
past few decades, because of both their scientific interest and management
implications Several factors that affect the cascading of effects down the trophic
chain have been noted.
Cascades in terrestrial food webs tend to be weaker than those of aquatic food
webs, perhaps because of greater food web reticulation in those cases, including
greater omnivory. The existence of diamond-shaped modules, a particular form of
reticulation in many food webs, is another important factor. Therefore, the study
of diamond-shaped webs is of special importance in understanding the
propagation of cascades.
Noonburg and Abrams (2005) showed that the violent transient dynamics resulting
from the indirect interactions of the two species might prevent the coexistence
from lasting in practice. However, intraspecific diamond-shaped food webs,
involving different phenotypes within a given consumer species, could be more
plausible.
M
N1
N2
R1
N3
R2
The above results can be
extended to some degree
both to more complex
models (e.g., to the left)
and to other functional
responses.
II. Intraspecific variation in habitat choice: with
loss during movement
This is the second topic. Here we explore some of the
consequences of two phenotypes of a consumer having traits
such that one phenotype uses one habitat, with resources and
predators, and the other uses another habitat, with a different
set of resources and predator. Unlike the case above, we now
assume there is energy loss or risk of mortality during
movement between habitats (patches).
Extending ESS to systems where there is loss during
movement
The Ideal Free Distribution (IFD) is commonly assumed for individuals of a
species distributed across habitat patches. The IFD does not take into
account that there can be losses in moving between habitat patches.
However, because many populations exhibit more or less continuous
population movement between patches, and travelling loss is a frequent
factor, it is important to determine the effects of losses on expected
population movement patterns and spatial distributions. It can be shown
that, if movement among patches is assumed, an evolutionarily stable
strategy (ESS) exists even when there are losses.
*Result of a NIMBioS workshop: Population and Community Ecology
Consequences of Intraspecific Niche Variation (Bolnick et al. PIs)
DeAngelis, D. L., Gail S. K. Wolkowicz, Yuan Lou, Yuexin Jiang, Mark Novak, Richard Svanback, Marcio
Araujo, YoungSeung Jo, and Erin Cleary. 2011. The effect of travel loss on evolutionary stable
distributions of populations in space. The American Naturalist 178:15-29.
We considered bitrophic chains in which the consumer can move freely and
continuously between two distinct patches with prey that are isolated in each
patch, and has perfect knowledge of the patches…
m12P1
(1-ε12)m12P1
P1
P2
(1-ε21) m21P2
a1R1P1
R1
Patch 1
Consumer, P
m21P2
a2R2P2
R2
Patch 2
Resource, R
… and we also considered tritrophic chains in which only
the consumer can move freely between patches.
M1
a1R1P1
R1
Patch 1
P2
Consumer, P
R2
Resource, R
(1-ε12)m12P1
P1
(1-ε21) m21P2
Predator, M
f2P2M2
f1P1M1
m12P1
M2
m21P2
a2R2P2
Patch 2
The equations for the two systems are as follows…
Bitrophic
Tritrophic
Tritrophic case
What we then did is assume
there is an invading
competing consumer, P1’ and
P2’.
Note that the migration rates
from patch 1 to patch 2 are
the same, m12, while the
migration rates from patch 2
to patch 1, m21 and m21’, differ
in general.
Otherwise the two competing
consumers are identical.
These equations can be solved
for equilibrium values.
Bitrophic case is analogous
We solved for the variables at equilibrium.
Equilibrium solution for tritrophic case
Note the bottom-up and top-down dependences are now the same as in the
classical tritrophic food chain
Finding an Evolutionarily Stable Strategy
Note that the equations for the invader and resident in patch 2 are the
following
It can be seen that the resident can exclude the invader by
appropriate choice of
and the invader can exclude the
resident by appropriate choice of . .
Equilibrium solution for bitrophic case
Similarly in this case, the resident or invader can choose a
movement rate back to Patch 1 that excludes the other
species.
So there is a value of m21 that the resident can choose such that it cannot
be successfully invaded by any possible competitor; or, conversely, an
invader using it that can replace any other strategy. Call it m21,opt.
Tritrophic case
Bitrophic case
This was shown to be an ESS. We can demonstrate numerically that a
resident with any m21 ≠ m21,opt , can be successfully invaded by an
invader that has m21’ = m21,opt (or that satisfies other conditions, see
later). Suppose the two patches are entirely identical (all parameters
are the same for the prey and consumers, in the bitrophic case).
Suppose also that the resident has m12 = 0.01m21,opt.
Then let an invader with m12 and with m21’ = m21,opt appear.
Bitrophic model simulations confirm that an invader with m12‘ at the
optimal value of m21,opt, starting from very small initial values, can
exclude any alternative resident strategy.
P1 and P2
Parameters
P1’ and P2’
R1 and R2
!
Note that although the loss rate, ε21, for returning to patch 1 is huge, ε21 = 0.99, the strategy
using the optimal return rate easily excludes the strategy using low return rate.
Selection for spatial asymmetry: bitrophic case
There are some
interesting properties of
this result. One is the
following. Suppose the
two patches are entirely
identical (all parameters
are the same for the
prey, predator, and
consumers).
Suppose also that the
resident has m12 = m21.
Let an invader with m21’
= m21,opt appear with
initially low numbers
(0.000001).
Invader P1’ and P2’
Resident P1 and P2
R1 and R2
Closer views of result, showing the emergence of asymmetry
P1 and P2 have identical values,
then start to decline
P1’ and P2’ increase from very
low values
R1 and R2 separate
P1’ and P2’ approach different
long-term values.
Result of asymmetry
This implies that the ESS for the distribution between identical patches is
spatially asymmetric.
This asymmetry emerges both in the movement coefficients m12 and m21 and
in the values of R1* and R2*, N1* and N2* , and M1* and M2*, even when the
parameters of the resources, consumers, and predators for the two
subpopulations are precisely the same. This is strange, because it means that
something also has to determine in which of the two subpopulation habitats
the resource, consumer, and predator take the values R1*, N1*, and M1*, and in
which these variables take the different values R2*, N2*, and M2*. This cannot
be answered from the analysis above.
Natural selection creates asymmetry in an initially homogeneous system when
there is loss in traveling.
Mathematical details can be
found in the appendices of
DeAngelis et al. (2011) and
in Lou and Wu (2011).
The latter includes proof of
the ESS for the tritrophic
case using a Lyapunov
function approach.
Implications in nature: Stream drift
The passive downstream drift caused by
one-directional flow of water is a
common pattern and Müller (1954,
1982) hypothesized that insects
compensate for downstream drift by a
tendency for the adult forms to fly
upstream to oviposit.
Implications in nature: Stream drift
Empirical studies have not conclusively supported the hypothesis that upstream movement
of adults compensates for the loss, but have shown that substantial degree of compensation
often occurs (Hershey et al. 1993 for mayflies)
Anholt (1995) proposed that such upstream movement may not be necessary, as density
dependence occurs in the aquatic stages of many insects, and drift of individuals from a
habitat patch may be compensated for by an increase in the survival rate of those remaining
on the patch.
Kopp et al. (2001), nevertheless, showed through invasion analysis simulation, that even in
such cases, upstream movement should be favored, because an insect genotype in which
losses to drift from upstream to downstream patches are exactly compensated for by
upstream movement will exclude any genotype for which this is not true.
Our results are more general than those of Kopp et al. (2001) and imply that if there are
losses in either direction or both, the optimal level of compensation through upstream
migration should be less than exact matching.
Further properties of ESS
When neither resident nor invader has the optimal
migration rate the results are more complex.
Resident only
m̂21
Coexistence
Invader only
m21,opt
Invader’s
migration rate
from patch 2
to patch 1
Invader only
Coexistence
Resident only
m21,opt
m21
Resident’s migration rate from patch 2 to patch 1
Note that if neither of the consumers has a travel rate exactly at m21,opt ,
then the sizes of the resident and invader populations is complex.
If both m12 and m12’ are on the same side of m21,opt , then only the population with closest rate to
m21,opt exists.
If m12 and m12’ are on opposite sides of m21,opt , then there is coexistence.
The results can be extended to a limited extend to N-patch systems
For a set of patches i = 1,N, the relevant equations for
the tritrophic case can be written as
It seems very difficult to get solutions for the resident and invader,
but a unique solution with the invader absent can be obtained.
An optimal movement rate can be obtained, but only under the
assumption that all rates, when obtained, are positive.
From the above it seems there are a lot of open
questions.
It should be again noted that we have made
several assumptions
•It is assumed there is movement from at least one patch,
which occurs from an ‘upstream’ patch.
•Populations are self-sustaining on every patch
•The different species or genotypes are identical in all
respects accept in their return rates to the first patch.
The above is only a small part of theory involving traveling with
loss. Much work relaxes the assumption of perfect knowledge.